Positive part and negative part of a real-valued function

In mathematics, the positive part and negative part of a real-valued function are two special functions assigned to this function. The positive part clearly agrees with the actual function if it assumes positive values ​​and is otherwise zero. The negative part of a function is defined in the same way.

definition

Positive part and negative part of the function outlined above

A real-valued function is given

${\displaystyle f\colon X\to \mathbb {R} }$.

Then the function is called with ${\displaystyle f^{+}\colon X\to \mathbb {R} }$

${\displaystyle f^{+}(x):=\max\{f(x),0\}={\begin{cases}f(x)&{\text{ falls }}f(x)\geq 0\\0&{\text{ sonst}}\end{cases}}}$

the positive part of and the function with ${\displaystyle f}$ ${\displaystyle f^{-}\colon X\to \mathbb {R} }$

${\displaystyle f^{-}(x):=-\min\{f(x),0\}={\begin{cases}-f(x)&{\text{ falls }}f(x)\leq 0\\0&{\text{ sonst}}\end{cases}}}$

the negative part of . It should be noted that there is also a positive function, i.e. it always applies to everyone . ${\displaystyle f}$ ${\displaystyle f^{-}}$ ${\displaystyle f^{-}(x)\geq 0}$ ${\displaystyle x}$

properties

It is

${\displaystyle f=f^{+}-f^{-}}$as well . ${\displaystyle |f|=f^{+}+f^{-}}$

Furthermore, a real-valued function can be measured precisely when its positive part and its negative part can be measured.

All of the above definitions or statements apply unchanged to numeric functions .

use

The positive part and the negative part of a function are often used in mathematical constructions. These are first defined for positive functions and then generalized to any functions by breaking down any functions into the positive part and the negative part (both of which are themselves positive functions).

A typical example of this is the Lebesgue integral : If the integral for positive measurable functions has been defined based on the simple functions , then the integral is defined over a measurable function (of any sign) as the integral over the positive part minus the integral over the negative part .

Web links

• Renze, John: Positive Part . In: MathWorld (English).
• Renze, John: Negative Part . In: MathWorld (English).

literature

• Achim Klenke: Probability Theory . 3. Edition. Springer-Verlag, Berlin Heidelberg 2013, ISBN 978-3-642-36017-6 , doi : 10.1007 / 978-3-642-36018-3 .
• Jürgen Elstrodt : Measure and integration theory . 6th, corrected edition. Springer-Verlag, Berlin Heidelberg 2009, ISBN 978-3-540-89727-9 , pp. 107 , doi : 10.1007 / 978-3-540-89728-6 .
• Norbert Kusolitsch: Measure and probability theory . An introduction. 2nd, revised and expanded edition. Springer-Verlag, Berlin Heidelberg 2014, ISBN 978-3-642-45386-1 , doi : 10.1007 / 978-3-642-45387-8 .

Individual evidence

1. ↑ Kusolitsch: Measure and probability theory. 2014, p. 86.
2. ↑ Klenke: Probability Theory. 2013, p. 91.